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180                        ANALYTICAL MECHANICS
done by the spider in climbing to the ceiling equals f mgk, where m is the mass of the spider and h its distance from the ceiling.
9.   The outer end of a flat spiral spring is fixed, while the inner end is attached to the center of a bar 20 cm. long, in such a way that the bar is parallel to the plane of the spring.   Two forces of 500 dynes each applied at the ends of the bar, at right angles to the bar and parallel to the plane
of the spring, can keep the bar turned through an angle of ^ radians.

What torque must be applied in order to keep the bar in position after giving it three turns?
10.   In the preceding problem find the work done in giving the bar three turns.   What portion of the total work is done in the last turn?
11.   Prove that the following is the expression for the torsional rigidity of a hollow shaft:
where b is the inner radius of the shaft, while the other letters represent the same magnitudes as in  147.
12.   Derive expressions for the saving of material and loss of rigidity due to making a shaft of a given external diameter hollow.
13.   Find the value of the quotient of the inner to the outer radius which will make the quotient of the saving of material to the loss of rigidity a maximum.
14.   The weight and the length of a shaft are fixed; find the ratio of the inner to the outer diameter which will make the rigidity of the shaft a maximum.
15.   The torsional moment which a shaft has to withstand and the length are fixed; find the ratio of the inner to the outer diameter which will make the weight of the shaft a minimum.
VIRTUAL WORK.
149. Principle of Virtual Work.  The concept of work enables us to formulate a principle, called the principle of virtual work, which can be applied to equilibrium problems to great advantage.
In order to derive this principle consider a particle which is in equilibrium. Evidently the resultant force acting upon the particle is nil and remains nil so long as the particle is in the equilibrium position. But when the particle is given a